When the Correct Model Fails: The Optimality of Stackelberg Equilibria with Follower Intention Updates

TL;DR

This paper analyzes a dynamic Stackelberg game where the leader updates beliefs about the follower's response, revealing that incorrect assumptions can sometimes lead to better outcomes for the leader.

Contribution

It characterizes the optimality of Stackelberg equilibria under belief updates about the follower's response, challenging the assumption that knowing the true response is always best.

Findings

Incorrect follower response assumptions can reduce leader costs.

Leader belief updates can improve outcomes in linear quadratic Stackelberg games.

Numerical simulations demonstrate non-trivial cases where wrong beliefs outperform correct ones.

Abstract

We study a two-player dynamic Stackelberg game where the follower's intention is unknown to the leader. Classical formulations of the Stackelberg equilibrium (SE) assume that the follower's best response (BR) function is known to the leader. However, this is not always true in practice. We study a setting in which the leader receives updated beliefs about the follower BR before the end of the game, such that the update prompts the leader and subsequently the follower to re-optimize their strategies. We characterize the optimality guarantees of the SE solutions under this belief update for both open loop and feedback information structures. Interestingly, we prove that in general, assuming an incorrect follower's BR may lead to a lower leader cost over the entire game than knowing the true follower's BR. We support these results with numerical examples in a linear quadratic (LQ) Stackelberg game, and use Monte Carlo simulations to show that the instances of incorrect BR achieving lower leader costs are non-trivial in collision avoidance LQ Stackelberg games.